Terzaghi consolidation: the series and its approximations

Terzaghi’s one-dimensional consolidation equation is the diffusion equation with a coefficient that happens to be called $c_v$:

For a layer of thickness $2H$ drained on both faces, with an excess pore pressure $u_0$ applied uniformly at $t = 0$, separation of variables gives the excess pore pressure as a sum of decaying modes, and integrating it over the layer gives the average degree of consolidation

in which $H$ is the longest path a water particle must travel to reach a drainage boundary, and

Everything else written about one-dimensional consolidation is either this series or an approximation to it. It is worth knowing which.

Every number in this article is recomputed from its inputs by a script that runs on each build of this site, including the series itself, which is summed rather than quoted.

The two approximationsPermalink to “The two approximations”#

The first is obtained by assuming the excess pore pressure distribution stays parabolic:

The second is a straight-line fit to the tail of the series on a semi-logarithmic plot:

Both are quoted, both are labelled “for $U$ below 60%” and “for $U$ above 60%” in the textbooks that are careful, and both are used outside those ranges by everybody else. The table shows what that costs.

Three things are worth reading off it.

The parabolic form is not an approximation at small $T_v$ — it is the answer. At $T_v = 0.05$ it agrees with the series to four decimal places, and at 0.10 the discrepancy is in the fourth. This is not luck. While the pressure front has not yet reached the centre of the layer, the problem is a half-space and the parabolic profile is the exact similarity solution.

It fails by exceeding certainty. At $T_v = 0.848$ it returns 103.9090%, and at 1.00 it returns 112.8379%. A formula that reports 112% consolidation is not slightly wrong; it has left the domain in which it means anything. That is a useful property, because it is self-announcing — unlike an approximation that quietly returns a plausible wrong number.

The logarithmic form is far better than its billing. At $T_v = 0.848$ it gives 90.0000% against the series’ 89.9979%, and even at $T_v = 0.30$, well below the 60% threshold it is usually restricted to, it is within 0.0083 of a percentage point. The restriction is conservative.

Where 0.197 and 0.848 come fromPermalink to “Where 0.197 and 0.848 come from”#

The two time factors everyone memorises are the inverses of the series at 50% and 90%. Solved on the series itself:

The parabolic form gives $T_{v,50} = \tfrac{\pi}{4}(0.5)^2 = 0.1963$, which is low by 0.194%. The logarithmic form gives $T_{v,90} = 1.781 - 0.933 = 0.8480$, low by 0.010%.

Neither error matters in a settlement calculation, because $c_v$ is known to a factor of two at best. Both matter in a verification calculation, where the question is not what the ground will do but whether the program solved the equation it was given. A finite element result compared against 0.1963 when the analytical answer is 0.1967 has been charged 0.2% of error that belongs to the reference, not to the code — and 0.2% is inside the band where several respectable codes actually live. This is the reason a benchmark must state which oracle it used at what precision, which is a theme of the KATAI 2D verification suite.

How many terms the series needsPermalink to “How many terms the series needs”#

The series is usually presented as if summing it were the hard part. It is not, but the effort is distributed in the opposite way to most people’s expectation. Terms needed for the neglected remainder to fall below $10^{-6}$:

Each term carries $\exp(-M^2 T_v)$ with $M$ growing linearly, so the exponent grows as the square of the term index. At $T_v = 0.5$ the second mode is already down by a factor of $e^{-9 \times 0.5 \times \pi^2/4}$ and the sum has converged. At small $T_v$ the damping is weak and many modes are alive at once — which is exactly the regime in which the parabolic form is exact, so the two methods are cheap in complementary places.

For a numerical implementation the consequence is a rule rather than a fixed term count: sum until the term is below tolerance, and do not hard-code ten terms because a table in a textbook used ten. A fixed count is the kind of choice that silently degrades near $T_v = 0.001$ and is never noticed, because nobody plots consolidation there.

What the time factor is actually forPermalink to “What the time factor is actually for”#

Take a 5 m clay layer, drained top and bottom, with $c_v = 2.4$ m²/yr. The drainage path is half the thickness:

Half the settlement in about six months, ninety percent in a little over two years, and the ratio $t_{90}/t_{50} = 4.31$ regardless of layer thickness or $c_v$ — it is a property of the series, not of the site.

Now remove one drainage boundary. The path doubles, $H^2$ quadruples, and

Four times as long, from a single change to a boundary condition. Sand lenses, working platforms, whether the underlying stratum drains — these decide the answer by a factor of four while $c_v$, over which laboratories and consultants argue, is squared into the denominator only once. It is the most leveraged assumption in the calculation and it is usually the least examined.

What the theory assumesPermalink to “What the theory assumes”#

The list is short and each item is violated routinely:

• Small strain. The drainage path is taken as constant, but it shortens as the layer compresses.
• Constant $c_v$. Both permeability and compressibility change with void ratio; they move in the same direction, which is why $c_v$ is more stable than either, and it is not why it is constant.
• Linear compressibility. $m_v$ is a secant, valid over the stress increment it was measured for and no other.
• No secondary compression. Creep continues after excess pore pressures have dissipated, and in an organic clay it can exceed the primary settlement.
• One dimension. Loads of finite width shed pore pressure sideways, which accelerates the process relative to this theory.

The last one is the reason coupled two-dimensional analyses exist, and the reason a code that reproduces this series exactly has demonstrated one thing about itself and not many. The stress increment that drives all of it is an effective stress increment, which is where this calculation begins rather than ends.